How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?
I want to parametrise so I can use the divergence theorem to calculate the flux along the surface above.
I don't know how to do it and would like ...

I'm having major trouble every time I need to parametrise a surface in order to take a surface integral, I just have no idea where to even start half of the time. Is there some kind of method that can ...

So since an elliptic surface with constant curvature would be a sphere, would a an hyperbolical surface with a constant curvature be the inside of a sphere if we were to go out from inside the sphere? ...

Let $S$ be the surface that is the graph of a continuous function $f: U \rightarrow \mathbb{R}$ on an open $U \subset \mathbb{R}^2$. Let $p = (x, y, f(x, y)) \in S$.
One usually defines the tangent ...

I know of the following formulas for calculating surface areas:
$\displaystyle A_S = 2\pi\int_{a}^{b}f(x)\sqrt{1+f'(x)^2}{\ dx}$ for the surface area ($A_S$) of the solid formed by revolving $f(x) = ...

I am currently learning about surfaces. So for the parametrized curve:
$r=\langle t^2, 3t\cos(2t), 3t\sin(2t)\rangle,\quad t\ge 1$
how can I find a equation for the surface the curve lie? Also what ...

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...

I want to calculate the surface of a body made of at least 3 overlapping ellipsoids. Below there is a picture of the cross section of the body.
I already know how to calculate the surface of single ...

Hi I'm looking for the equation of a cheese twist in 3d (either parametric or cartesian)... Can be multiple planes but was wondering if anyone had any idea to execute something like this? Thanks
e.g.
...

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates.
I define a deformed sphere ...

What is the difference between a conic and a quadric? I'm guessing that this depends on your ambient space? I think that conics are just special quadrics and are a codimension 1 object and a quadric ...

I want to prove that given a generalized cylinder $C(s,t)=\alpha(s)+t\hat{z}$ , where $\alpha$ is a curve on the $xy$ plane and $\hat{z}$ is the $z$-axis vector, then a geodesic curve $\gamma$ has the ...

Suppose two curves $\gamma$ and $\gamma'$ are diffeomorphic.
Is the arc-length measure $ds_\gamma$ absolutely continuous to $ds_\gamma'$ with a positive derivative? ($ds_\gamma=\phi\, ds_\gamma'$ for ...

I'm interested in the structure of homeo- and diffeomorphism groups of oriented surfaces, especially in hyperbolic case. For example, does the homeomorphism group retracts on the diffeomorphism group ...

In Lee's book Introduction to Topological Manifolds, he discusses polygonal presentations of surfaces. He does so by means of words $W_1, \dotsc, W_n$ such that each letter that appears must appear ...

Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$.
Definition
A surface in $\Bbb R^3$ is a subset $M$ of $R^3$ such that for each point $p$ of $M$ there exists a ...

I'm reading some notes on hyperbolic surfaces by François Labourie and there's an exercise I can't figure out.
I have to prove that the length l(c) of a curve does not depend on the subdivision. It's ...

the part of the sphere given by:
$$ S = \{ (x,y,z) | x^2+y^2+z^2 = 25, -4 \leq x,y,z \leq 4 \} $$
first Q: I'm not sure if I can apply to this Divergence theorem ? It seem that in order to use it I ...

Problem
Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries open sets in $E$ to open sets in $M$. Deduce that if $\mathbf{x}:D \to M$ is an arbitrary patch, then the image ...

Let us consider a surface $S$ and a subset $T\subset S$, where $T\cong D^2$ are homeomorphic and $D^2$ is an open disc.
Let $T\cong Q\subset S$ be homeomorphic.
Are $S\setminus T $ and $S\setminus Q$ ...